1. Linear Regression and Correlation Analysis

2. Methods for Studying Relationships
Chapter Goals
To understand the methods for displaying and describing relationship among two variables
Methods for Studying Relationships
Models
Linear regression
Correlations
Frequency tables
Graphical
Scatter plots
Line plots
3-D plots

3. Two Quantitative Variables
The response variable, also called the dependent variable, is the variable we want to predict, and is usually denoted by y.
The explanatory variable, also called the independent variable, is the variable that attempts to explain the response, and is denoted by x.

4. Scatter Plots and Correlation
A scatter plot (or scatter diagram) is used to show the relationship between two variables
Correlation analysis is used to measure strength of the association (linear relationship) between two variables
Only concerned with strength of the relationship
No causal effect is implied

5. Example
The following graph shows the scatter plot of Exam 1 score (x) and Exam 2 score (y) for 354 students in a class.
Is there a relationship between x and y?

6. Scatter Plot Examples y x Linear relationships
Curvilinear relationships

7. Scatter Plot Examples
(continued)
No relationship y x y x

8. Correlation Coefficient
(continued)
The population correlation coefficient ρ (rho) measures the strength of the association between the variables
The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations

9. Features of ρ and r Unit free Range between -1 and 1
The closer to -1, the stronger the negative linear relationship
The closer to 1, the stronger the positive linear relationship
The closer to 0, the weaker the linear relationship

10. Examples of Approximate r Values
Tag with appropriate value: -1, -.6, 0, +.3, 1 y x y

11. Earlier Example

12. Questions?
What kind of relationship would you expect in the following situations:
Age (in years) of a car, and its price.
Number of calories consumed per day and weight.
Height and IQ of a person.

13. Exercise
Identify the two variables that vary and decide which should be the independent variable and which should be the dependent variable. Sketch a graph that you think best represents the relationship between the two variables.
The size of a persons vocabulary over his or her lifetime.
2. The distance from the ceiling to the tip of the minute hand of a clock hung on the wall.

14. Introduction to Regression Analysis
Regression analysis is used to:
Predict the value of a dependent variable based on the value of at least one independent variable.
Explain the impact of changes in an independent variable on the dependent variable.
Dependent variable: the variable we wish to explain.
Independent variable: the variable used to explain the dependent variable.

15. Simple Linear Regression Model
Only one independent variable, x.
Relationship between x and y is described by a linear function.
Changes in y are assumed to be caused by changes in x.

16. Types of Regression Models
Positive Linear Relationship
Negative Linear Relationship
Relationship NOT Linear
No Relationship

17. Population Linear Regression
The population regression model:
Population y intercept
Population Slope Coefficient
Random Error term, or residual
Independent Variable
Dependent Variable
Random Error
component
Linear component

18. Linear Regression Assumptions
The Error terms εi, i=1, 2. …, n are independent and εi ~ Normal (0, σ2).
The Error terms εi, i=1, 2. …, n have constant variance σ2.
The underlying relationship between the X variable and the Y variable is linear.

19. Population Linear Regression
(continued) Y
Observed Value of yi for xi εi
Slope = β1
Predicted Value of yi for xi
Random Error for this xi value
Y-Intercept = β0 xi X

20. Estimated Regression Model
The Regression Function:
The sample regression line provides an estimate of the population regression function.
Estimate of the regression y-intercept
Estimate of the regression slope
Estimated (or predicted) y value
Independent variable
The individual random error terms ei have a mean of zero

21. Earlier Example

22. Least Squares Criterion
Residual
A residual is the difference between the observed response yi and the predicted response ŷi. Thus, for each pair of observations (xi, yi), the ith residual is ei = yi − ŷi = yi − (b0 + b1xi)
Least Squares Criterion
b0 and b1 are obtained by finding the values of b0 and b1 that minimize the sum of the squared residuals.

23. Interpretation of the Slope and the Intercept
b0 is the estimated average value of y when the value of x is zero.
b1 is the estimated change in the average value of y as a result of a one-unit change in x.

24. The Least Squares Equation
The formulas for b1 and b0 are:
algebraic equivalent:
and

25. Finding the Least Squares Equation
The coefficients b0 and b1 will usually be found using computer software, such as Excel, Minitab, or SPSS.
Other regression measures will also be computed as part of computer-based regression analysis.

26. Simple Linear Regression Example
A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)
A random sample of 10 houses is selected
Dependent variable (y) = house price in $1000s
Independent variable (x) = square feet

27. Sample Data for House Price Model
House Price in $1000s
(y)
Square Feet
(x)
245
1400
312
1600
279
1700
308
1875
199
1100
219
1550
405
2350
324
2450
319
1425
255

28. SPSS Output
The regression equation is:

29. Graphical Presentation
House price model: scatter plot and regression line
Slope
= 0.110
Intercept =

30. Interpretation of the Intercept, b0
b0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values)
Here, no houses had 0 square feet, so
b0 = just indicates that, for houses within the range of sizes observed, $98, is the portion of the house price not explained by square feet

31. Interpretation of the Slope Coefficient, b1
b1 measures the estimated change in the average value of Y as a result of a one-unit change in X
Here, b1 = tells us that the average value of a house increases by ($1000) = $109.77, on average, for each additional one square foot of size

32. Least Squares Regression Properties
The sum of the residuals from the least squares regression line is 0 ( ).
The sum of the squared residuals is a minimum (minimized ).
The simple regression line always passes through the mean of the y variable and the mean of the x variable.
The least squares coefficients b0 and b1 are unbiased estimates of β0 and β1.

33. Exercise
The growth of children from early childhood through adolescence generally follows a linear pattern. Data on the heights of female Americans during childhood, from four to nine years old, were compiled and the least squares regression line was obtained as ŷ = x where ŷ is the predicted height in inches, and x is age in years.
Interpret the value of the estimated slope b1 = 2. 4.
Would interpretation of the value of the estimated y-intercept, b0 = 32, make sense here?
What would you predict the height to be for a female American at 8 years old?
What would you predict the height to be for a female American at 25 years old? How does the quality of this answer compare to the previous question?

34. Coefficient of Determination, R2
The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable.
The coefficient of determination is also called R-squared and is denoted as R2

35. Coefficient of Determination, R2
(continued)
Note: In the single independent variable case, the coefficient of determination is
where:
R2 = Coefficient of determination
r = Simple correlation coefficient

36. Examples of Approximate R2 Valuesy y x x y y x x

37. Examples of Approximate R2 Valuesy
No linear relationship between x and y:
The value of Y does not depend on x. (None of the variation in y is explained by variation in x) x
R2 = 0

38. SPSS Output

39. Standard Error of Estimate
The standard deviation of the variation of observations around the regression line is called the standard error of estimate
The standard error of the regression slope coefficient (b1) is given by sb1

40. SPSS Output

41. Comparing Standard Errors
Variation of observed y values from the regression line
Variation in the slope of regression lines from different possible samples y y x x y y x x

42. Inference about the Slope: t-Test
t-test for a population slope
Is there a linear relationship between X and Y?
Null and alternate hypotheses
H0: β1 = 0 (no linear relationship)
H1: β1  0 (linear relationship does exist)
Test statistic:
Degree of Freedom:
where:
b1 = Sample regression slope
coefficient
β1 = Hypothesized slope
sb1 = Estimator of the standard
error of the slope

43. Example: Inference about the Slope: t Test
(continued)
Estimated Regression Equation:
House Price in $1000s
(y)
Square Feet
(x)
245
1400
312
1600
279
1700
308
1875
199
1100
219
1550
405
2350
324
2450
319
1425
255
The slope of this model is
Does square footage of the house affect its sales price?

44. Inferences about the Slope: t Test Example - Continue
Test Statistic: t = 3.329 b1 t*
H0: β1 = 0
HA: β1  0
From Excel output:
Coefficients
Standard Error
t Stat
P-value
Intercept
Square Feet
d.f. = 10-2 = 8
Decision: Reject H0
Conclusion:
a/2=.025
a/2=.025
There is sufficient evidence that square footage affects house price
Reject H0
Do not reject H0
Reject H0
-tα/2
t(1-α/2)
2.3060
3.329

45. Confidence Interval for the Slope
Confidence Interval Estimate of the Slope:
d.f. = n - 2
Excel Printout for House Prices:
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Square Feet
At 95% level of confidence, the confidence interval for the slope is (0.0337, )

46. Confidence Interval for the Slope
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
Square Feet
Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $ per square foot of house size
This 95% confidence interval does not include 0.
Conclusion: There is a significant relationship between house price and square feet at the .05 level of significance

47. Residual Analysis Purposes Examine for linearity assumption
Examine for constant variance for all levels of x
Evaluate normal distribution assumption
Graphical Analysis of Residuals
Can plot residuals vs. x
Can create histogram of residuals to check for normality

48. Residual Analysis for Linearityx x x x
residuals
residuals 
Not Linear
Linear

49. Residual Analysis for Constant Variance
Non-constant variance 
Constant variance x y
residuals

50. Example: Residual Output
Predicted House Price
Residuals 1 2 3 4 5 6 7 8 9 10